What are the weaknesses of the Ptolemaic geocentric model that were pointed out by the Copernican theory at the time the Heliocentric model was first presented?
Since Copernicus died after publishing the Heliocentric model, it was his followers and not Copernicus himself who would have pointed out the Ptolemaic weaknesses.
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In terms of mathematics, both the Ptolemaic model and the Copernican model come out to be the same. Both make the exact same planetary motion predictions and both use the same geometry. Therefore, the weaknesses in both the Ptolemaic model and the Copernican are exactly the same; neither is weaker than the other. However, it is believed that Copernicus devised his theory because he did not like the way Ptolemy compromised Aristotle's idea of planetary uniform circular motion by constructing the equant. He also viewed other abuses of Aristotle's professed uniform circular motion, which led Copernicus to conclude that a heliocentric model is the most accurate. Followers of Copernicus would have also objected to Ptolemy's deviation from uniform motion as a weakness in Ptolemy's model ("Nicolaus Copernicus").
Ptolemy observed, contrary to Aristotle's profession, that the motions of the planets are actually not uniform. In order to get around this idea, he constructed a model that allowed for more complex movement that was still geometrically Aristotelian in that the planets still moved in circles. The planets not only moved in one circle around the earth, called the deferent, they also moved in smaller circles relative to each planet, called epicycles. Picture smaller wheels on a circular mechanism, with one point on each wheel representing the planet, spinning as the larger circle rotates. Copernicus's model uses similar motions; however, the part about Ptolemy's model that really bothered Copernicus was the use of the equant. The equant was a line of motion off-center from the circle around which the planets moved. It actually accounts for the elliptical movement of the planets in our model today (Fitzpatrick, "Ptolemy's Model"). However, since this line of motion is not moving in uniform circles, but rather in an elliptical motion, Copernicus wanted to construct a model that did away with the need for the equant.
Removing the equant, while still accounting for the non-uniform motion, meant for Copernicus that the non-uniform planetary motion can now be explained as the earth's and other planet's movements around the sun rather than the sun's movements. While still keeping the use of the epicycles and deferents, but adjusting the size and speed of the planets' orbits, Copernicus was able to create a model consisting entirely of completely uniform circles of the same size, which he felt was the more Aristotelian model (Fitzpatrick, "Copernicus's Model"). Copernicus and his followers would argue that Ptolemy's lack of uniform circles is a failing in Ptolemy's model.
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